The importance of the concept of stability in fractional order system and control has been recognized for some time now. Recently, it has become evident that many conclusions were drawn, but little consensus was reached. Consequently, there is an urgent need for a much deeper understanding of such a concept. With the definition of fractional order positive definite matrix, a set of equivalent and elegant stability criteria are developed via revisiting a stability criterion we proposed before. All the results are formed in terms of linear matrix inequalities. Afterwards, a series of interesting properties of these criteria are revealed profoundly, including completeness, singularity, conservatism, etc. Eventually, a simulation study is provided to validate the effectiveness of the obtained results.
In this paper, from the classical short memory principle under Grünwald-Letnikov definition, several novel short memory principles are presented and investigated. On one hand, the classical principle is extended to Riemann-Liouville and Caputo cases. On the other hand, a special kind of principles are formulated by introducing a discrete argument instead of the continuous time, resulting in principles with fixed memory length or fixed memory step. Apart from these, several interesting properties of the proposed principles are revealed profoundly.
In this paper, a generalized fractional central difference Kalman filter for nonlinear discrete fractional dynamic systems is proposed. Based on the Stirling interpolation formula, the presented algorithm can be implemented as no derivatives are needed. Besides, in order to estimate the state with unknown prior information, a maximum a posteriori principle based adaptive fractional central difference Kalman filter is derived. The adaptive algorithm can estimate the noise statistics and system state simultaneously. The unbiasedness of the proposed algorithm is analyzed. Several numerical examples demonstrate the accuracy and effectiveness of the two Kalman filters.
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